Why is the last layer of a DBN or DBM used for classification task?
One of the big realizations that deep learning models brought in recent years was that we can train the feature extractors and classifiers simultaneously. In fact most people have stopped separating the 2 tasks and simply refer to all the process as training the model. 

However, if you dive in to every single model architecture, it will always be constructed from the first part which is the feature extractor which outputs the embedding output - (which is basically the x encoded features of the input), and second part consisting of the final layer the model - the classifier which uses the embedding layer encoding to predict the class of the input.

The goal of the first part is to reduce the dimensionality of the input to just the most impotent features for the final task. The goal of the classifier is to use those features to output the final score/class etc.

This is why usually only this layer is fine-tuned, because we don't want to damage the trained feature extractor, just update the classifier to fit a slightly different distribution.

I'm pretty sure that in your mentioned case, for generation they do not use the classification layer, so updating it shouldn't have any affect on the model's generative abilities.

Regarding your last question, yes it is possible, ones you extracted the features with the model, you can use any kind of classifier on them.